Why Is "dishwash soap" incorrect?
In British English it is usually called washing up liquid, while in the US it is usually called dish soap.
I wouldn't say it is wrong, but it's unidiomatic. I've never heard native speakers say that..
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Hi Clive.... hi Jay!

How about "dish - washing liquid" or "dish - washing soap"?
LaboriousHi Clive.... hi Jay!How about "dish - washing liquid" or "dish - washing soap"?
Both are possible, although less common than the two I mentioned earlier.
I say dish-washing liquid,

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LaboriousHi Clive.... hi Jay!How about "dish - washing liquid" or "dish - washing soap"?
They are unnecessary because, as Blue Jay already mentioned, there are established terms for it in British English and the USA.

You could use your suggestions if someone did not know the established terms and you therefore had to describe what you were talking about.

Chemically, these products are hardly ever made of actual soap.

Up until at least the 1970s, these products were sold as "detergent" -- "dish detergent", "dishwashing detergent", or "dishwashing liquid". However, colloquial usage was to call detergents "soap" if they were at all soaplike. In the middle 1970s, the US Federal Trade Commission released an opinion that products could be labeled as "soap" if that's what the customer would think of it as. So first liquid products for washing hands started being labeled as soap even if they contained no soap, and then eventually liquids for hand dishwashing started to be labeled as "dish soap" or "dishwashing soap" as well.

This actually helped resolve some confusion as the products for washing by hand could be labeled as "soap", and the term "detergent" reserved for those for washing by machine. There is some sense to that distinction, because the compositions for machine dishwashing are very un-soaplike, while detergents for washing dishes by hand are still soaplike even if they contain no soap. Today young people look at the old labels of Ivory Liquid Gentle White Detergent and wonder what people must have used to wash dishes then, as I've been told "detergent" now has the connotation of something abrasive!

Unfortunately brevity beats all, as people continue to refer to detergent to put in the dishwashing machine as "soap", much to the consternation of those who then follow that colloquial instruction and wind up with a foamy kitchen.

If you really want to be correct and clear, at the cost of sounding pedantic, "hand dishwashing detergent" and "machine dishwashing detergent" are unambiguous. If you want to specify actual soap, then the qualifier "castile" may work in most cases, unfortunately further burying the original distinction of castile soap as olive oil soap; not wanting to do so, I find myself writing "actual soap soap".

In Britain the use of the noun phrase "the washing-up", equivalent to the French "la vaisselle", leads to products for hand dishwashing to be called "washing-up liquid" or "washing-up fluid".